To create a square root function that models the given data, we will need to define a mathematical relationship between the cake area and the cake pan radius. Given the nature of the data, a square root function can be expressed in the form:
\[ r(A) = k \sqrt{A} \]
where:
- \( r \) is the radius of the cake pan,
- \( A \) is the area of the cake,
- \( k \) is a constant to be determined.
After rearranging the equation, we have:
\[ \frac{r(A)}{\sqrt{A}} = k \]
We can calculate \( k \) for each point from the data set to find a suitable model. Here’s the data provided:
| Cake Area (in\(^2\)) | Cake Pan Radius (in.) | |-----------------------|------------------------| | 0 | 0 | | 25 | 2.8 | | 40 | 3.54 | | 55 | 4.15 | | 70 | 4.69 |
Step 1: Calculate \( k \)
Let's calculate \( k \) for the non-zero areas:
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For \( A = 25 \): \[ k = \frac{2.8}{\sqrt{25}} = \frac{2.8}{5} = 0.56 \]
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For \( A = 40 \): \[ k = \frac{3.54}{\sqrt{40}} \approx \frac{3.54}{6.32} \approx 0.56 \]
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For \( A = 55 \): \[ k = \frac{4.15}{\sqrt{55}} \approx \frac{4.15}{7.42} \approx 0.56 \]
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For \( A = 70 \): \[ k = \frac{4.69}{\sqrt{70}} \approx \frac{4.69}{8.37} \approx 0.56 \]
Step 2: Confirm the value of \( k \)
Across the measurements, it appears that \( k \approx 0.56 \) consistently. Thus, we can say that the square root function that models this data is:
\[ r(A) \approx 0.56 \sqrt{A} \]
Final Model:
To summarize, the final function that models the radius of the cake pan as a function of the cake's area is:
\[ r(A) \approx 0.56 \sqrt{A} \]
Rounding:
Since the problem requests rounding to the nearest hundredth, we find:
\[ k \approx 0.56 \]
Thus, the rounded model function remains:
\[ r(A) \approx 0.56 \sqrt{A} \]
This function can now be used to estimate the cake pan radius for any cake area within the observed range.